How to know if a MacLaurin/Taylor Series expansion is good? This question is motivated by this question.
So, given $\frac{1}{e^x + 1}$, the 4th order MacLaurin series $1 -e^x+(e^x)^2-(e^x)^3+(e^x)^4$, although correct in terms of the algebra manipulations, is not a good expansion.
In general, how do we know if a given expansion is a good approximation? Should we be extra careful when the given function is a fraction and has a variable in its denominator or are there other cases where we should pause and see how the given expansion behaves with small values of $x$?
 A: The expansion you listed will be good when $e^x$ is "small" (close to zero). Since $e^0=1$ the approximation will be very bad near the origin. 
All taylor series have a radius of convergence. That is a restricted range of values for $x$ which you can plug in and expect to get a good answer. In the case of the geometric series the range of values for $x$ is $(-1,1)$ and if $x$ is close to zero you won't need many terms for a good approximation.
$$ \frac{1}{1-x} = 1+x+x^2+\cdots \qquad (\vert x \vert < 1)$$
Every Taylor series has some point about which you are expanding. In the case of Mclaurin series that point is $x_0$=0. The closer the series is to this point the better the approximation. Taylor's theorem provides an explicit upper bound on the error for a truncated series. For instance taking the 5th order taylor approximation of a function about a point $x_0$ will give an error R which is bounded by,
$$ R \leq M(x-x_0)^6,$$
where $M$ is a constant.
See Taylor Series Remainder for more.

Thought I would add this:
When you're putting a function into a taylor series you just have to make sure its range is in the interval of convergence for the series. For instance if you plugged $sin(x)$ into your series it would converge very nicely near the origin and then get screwed up when $sin$ gets close to $1$ or $-1$ its the numerical value of the function which is relevant.
A: I'm guessing you want to then expand in the relevant series for $e^{x}$, $e^{2x}$, etc. as in your other question (getting ultimately something like $\frac{95}{12} \, x^{4} + \frac{22}{3} \, x^{3} + 5 \, x^{2} + 2 \, x + 1$).
To see what's going wrong, look at the radius of convergence for the first series you want to use, i.e. $\dfrac{1}{1+x} = 1 - x + x^2 - x^3 + x^4 + O(x^5)$. The radius of convergence is $|x| < 1$, so if you want to use this series to calculate $\dfrac{1}{1 + f(x)}$, then $1 - f(x) + (f(x))^2 - (f(x))^3 \dots$ is only a valid expansion if $|f(x)| < 1$. 
If you want to then use a further Taylor series expansion for $f(x)$, you have to worry about the radius of convergence for that series as well (though in this case, $e^x$ converges everywhere, so we're off the hook there).
Thus, your series is only good if $|e^x| = e^x < 1$, so for $x<0$. If you want a series expansion for other values of $x$, you'll have to do something else (e.g. follow the suggestion in your other question, or just 'start from scratch' by taking derivates, etc.).
